Nuclear Locally Convex Spaces [Reprint 2021 ed.] 9783112564103, 9783112564097

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Albrecht Pietsch N U C L E A R LOCALLY C O N V E X S P A C E S

Albrecht Pietsch

Nuclear Locally Convex Spaces Translated from the Second German Edition by William H. Ruckle .

A K A D E M I E - V E R L A G 19

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B E R L I N

Translation of the German edition: Nukleare lokalkonvexe R ä u m e , 2 . A u f l a g e Copyright 1969 b y Akademie-Verlag G m b H , Berlin

Lizenzausgabc vom Akademie-Verlag GmbH, 108 Berlin, Leipziger Straße 3 — 4 (c) by Springer-Verlag Berlin» Heidelberg 1972 Lizenznummer: 202 • 100/560/72 Herstellung: VEB Druckerei,,Thomas Müntzer", 582 Bad Langensalza Bestellnummer: 761 691 0 (5990) • ES 19 B 4 Printed in German Democratic Republic

Foreword to the First Edition

With a few exceptions the locally convex spaces encountered in analysis can be divided into two classes. First, there are the normed spaces, which belong to classical functional analysis, and whose theory can be considered essentially closed. The second class consists of the so-called nuclear locally convex spaces, which were introduced in 1951 b y A. Grothendieck. The two classes have a trivial intersection, since it can be shown t h a t only finite dimensional locally convex spaces are simultaneously normable and nuclear. It can be asserted without exaggeration that the most important part of the theory of nuclear locally convex spaces is already contained in the fundamental dissertation of A. Grothendieck. Unfortunately, the machinery of locally convex tensor products which is used there is very cumbersome, so t h a t many proofs are unnecessarily complicated. Thus, in this book I have undertaken the task of constructing the theory of nuclear locally convex spaces without using locally convex tensor products. Their place is taken by locally convex spaces of summable and absolutely summable families, which are introduced in the first chapter. The great advantage of this method is not merely the simplicity of the proofs. Indeed, the most important result is a necessary and sufficient condition for the strong topological dual of a nuclear locally convex space to be nuclear, which appears here for the first time. In recent years certain concepts of approximation theory have attained great significance in the treatment of nuclear locally convex spaces. This statement is especially applicable to the concept of approximative dimension introduced by A. N. Kolmogorov and A. Pelczynski and applied by B. S. Mitiagin to characterize nuclear locally convex spaces. The principle results of these recent investigations are described in the three final chapters. We single out the Basis Theorem which has great significance for the representation theory of nuclear locally convex spaces. The main conclusion from these studies can be summarized in the following statement: If a theory of structure for locally convex spaces can be developed at all, then it must certainly be possible for nuclear locally convex spaces because these are more

VI

Foreword

closely related to finite dimensional locally convex spaces than are normed spaces. In order to present a clear narrative I have omitted exact references to the literature for individual propositions. However, each chapter begins with a short introduction which also contains historical remarks. Deutsche Akademie der Wissenschaften zu Berlin Institut für Reine Mathematik Albrecht Pietsch

Foreword to the Second Edition

Since the appearance of the first edition, some important advances have taken place in the theory of nuclear locally convex spaces. Firsts there is the Universality Theorem of T. and Y . Kömura which fully confirms a conjecture of Grothendieck. Also, of particular interest are some new existence theorems for bases in special nuclear locally convex spaces. Recently many authors have dealt with nuclear spaces of functions and distributions. Moreover, further classes of operators have been found which take the place of nuclear or absolutely summing operators in the theory of nuclear locally convex spaces. Unfortunately, there seem to be no new results on diametral or approximative dimension and isomorphism of nuclear locally convex spaces. Since major changes have not been absolutely necessary I have restricted myself to minor additions. Only the tenth chapter has been substantially altered. Since the universality results no longer depend on the existence of a basis it was necessary to introduce an independent eleventh chapter on universal nuclear locally convex spaces. In the same chapter s-nuclear locally convex spaces are also briefly treated. I have brought the bibliography up to date as well as I could. The newly added table of symbols ought to make the reader's job easier. Friedrich-Schiller-Universität, Jena Mathematisch-Naturwissenschaftliche Fakultät

Albrecht Pietsch

Contents

Chapter o. Foundations 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. 0.8. 0.9. 0.10. 0.11. 0.12.

Topological Spaces Metric Spaces Linear Spaces Semi-Norms Locally Convex Spaces The Topological Dual of a Locally Convex Space Special Locally Convex Spaces Banach Spaces Hilbert Spaces Continuous Linear Mappings in Locally Convex Spaces The Normed Spaces Associated 'with a Locally Convex Space Radon Measures

Chapter 1. Summable Families 1.1. 1.2. 1.3. 1.4. 1.5. 1.6.

Summable Families of Numbers Weakly Summable Families in Locally Convex Spaces Summable Families in Locally Convex Spaces Absolutely Summable Families in Locally Convex Spaces Totally Summable Families in Locally Convex Spaces Finite Dimensional Families in Locally Convex Spaces

Chapter 2. Absolutely Summing Mappings 2.1. 2.2. 2.3. 2.4. 2.5.

Absolutely Summing Mappings in Locally Convex Spaces Absolutely Summing Mappings in Normed Spaces A Characterization of Absolutely Summing Mappings in Normed Spaces A Special Absolutely Summing Mappings Hilbert-Schmidt Mappings

Chapter 3. Nuclear Mappings 3.1. 3.2. 3.3. 3.4.

Nuclear Mappings in Normed Spaces Quasinuclear Mappings in Normed Spaces Products of Quasinuclear and Absolutely Summing Mappings in Normed Spaces The Theorem of Dvoretzky and Rogers

Chapter 4. Nuclear Locally Convex Spaces 4.1. 4.2. 4.3. 4.4.

Definition of Nuclear Locally Convex Spaces Summable Families in Nuclear Locally Convex Spaces The Topological Dual of Nuclear Locally Convex Spaces Properties of Nuclear Locally Convex Spaces

1 1 3 4 5 6 S 10 11 11 13 14 16 18 18 23 25 27 29 32 34 34 36 38 42 45 49 49 55 60 67 69 69 72 76 79

Contents Chapter 5. P e r m a n e n c e Properties of Nuclearity 5.1. 5.2. 5.3. 5.4. 5.5.

Subspaces a n d Q u o t i e n t Spaces Topological P r o d u c t s a n d S u m s Complete H u l l s Locally Convex Tensor P r o d u c t s Spaces of Continuous Linear Mappings

Chapter 6. Examples of Nuclear Locally Convex Spaces 6.1. 6.2. 6.3. 6.4.

Sequence Spaces of Spaces of Spaces of

Spaces I n f i n i t e l y Differentiable F u n c t i o n s Harmonic Functions Analytic Functions

Chapter 7. Locally Convex Tensor Products 7.1. 7.2. 7.3. 7.4. 7.5.

Definition of Locally Convex Tensor P r o d u c t s Special Locally Convex Tensor P r o d u c t s A Characterization of Nuclear Locally Convex Spaces The Kernel Theorem The Complete jr-Tensor P r o d u c t of N o r m e d Spaces

Chapter 8. Operators of Type ft a n d s 8.1. 8.2. 8.3. 8.4. 8.5. 8.6.

83 85 90 93 94 95 97 97 99 102 105 107 108 109 113 115 117 120

The A p p r o x i m a t i o n N u m b e r s of Continuous L i n e a r M a p p i n g s in N o r m e d Spaces 120 Mappings of T y p e ft 125 The A p p r o x i m a t i o n N u m b e r s of Compact Mappings in H i l b e r t Spaces . 129 Nuclear a n d Absolutely S u m m i n g Mappings 135 M a p p i n g s of T y p e s 138 A Characterization of Nuclear Locally Convex Spaces 141

Chapter 9. D i a m e t r a l a n d Approximative Dimension 9.1. 9.2. 9o9.4. 9.5. 9.6. 9.7. 9.8.

IX

T h e D i a m e t e r of B o u n d e d Subsets in N o r m e d Spaces The D i a m e t r a l Dimension of Locally Convex Spaces The D i a m e t r a l Dimension of P o w e r Series Spaces The D i a m e t r a l Dimension of Nuclear Locally Convex Spaces . . . A Characterization of D u a l Nuclear Locally Convex Spaces The £ - E n t r o p y of B o u n d e d Subsets in N o r m e d Spaces The A p p r o x i m a t i v e Dimension of Locally Convex Spaces The A p p r o x i m a t i v e Dimension of Nuclear Locally Convex Spaces . .

144 144 149 151 .155 157 160 164 .167

Chapter 10. Nuclear Locally Convex Spaces with Basis

171

10.1. 10.2. 10.3-

1 72 173 175

Locally Convex Spaces w i t h Basis R e p r e s e n t a t i o n of Nuclear Locally Convex Spaces w i t h Basis Bases in Special Nuclear Localty Convex Spaces

Chapter 11. Universal Nuclear Locally Convex Spaces

177

I m b e d d i n g in t h e P r o d u c t Space ( 2 J ) 1 I m b e d d i n g in t h e P r o d u c t Space (JF")J

177 179

11.1. 11.2.

Bibliography

181

Index

190

Table of Symbols

193

Chapter 0

Foundations

This chapter presents the concepts and propositions which we shall use. Besides the most important definitions and a few elementary statements, all that we shall need from general topology is the fundamental Tichonov Theorem. The real foundation of our investigation is the theory of locally convex spaces. In addition to the Hahn Banach Theorem, we shall also need some other deep theorems. On the other hand, only some elementary statements on Hilbert space are necessary. This is also true for measure theory from which we take only the definition of Radon measure on a compact Hausdorff space and the construction of the Hilbert space £?(M). References to these areas are: Topology: N. Bourbaki [1], J . L. Kelley [1], G. Kothe [4]. Locally Convex Spaces: N. Bourbaki [4], A. Grothendieck [8], G. Kothe [4], J . L. Kelley and I. Namioka [1], H. H. Schaefer [1]. Normed Spaces: S. Banach [1], M. M. Day [1]. Hilbert Spaces: N. I. Ahiezer and I. H. Glazman [1], P. Halmos [2]. Measure Theory: N. Bourbaki [3], P. Halmos [1]. 0.1. Topological Spaces 0.1.1. A topology X is given on a set M = {x, y, . . .} if to each element x € M there correspondents a non-empty system of subsets which satisfy the following conditions: (E/j) If U e U5.(*) then x e U. (U2) For finitely many sets Uly . . . , Un € there is a set U e U g (*) with U c U1 n • • • n Un. (U3) Each set U e contains a subset U' 6 such that for each element y e U' there is a set V € with V c U. A set with a topology is called a topological space. 0.1.2. A subset U of a topological space M is called a neighborhood of the element * if there is a set U0 in U~(%) with U0 c U. Ug(^) is then referred to as a fundamental system of neighborhoods of the element x.

2

0. Foundations

0.1.3. A topology is finer than a topology if each ^-neighborhood of an arbitrary element x € M is also a ^-neighborhood. The topology is then said to be coarser than the topology Two topologies on a set -M are equal if all elements x e M have the same neighborhoods. 0.1.4. A topology can be introduced on each subset M' of a topological space M by taking the collection of intersections U' = M' n U with U £ as a fundamental system of neighborhoods for each element ^ e M'. This is called the topology induced on M' by M. 0.1.5. A directed system {xx} in a topological space M is a set of elements xx e M which are uniquely associated with the elements of an index set A. For certain pairs of indices tx and in A a relation tx S; /? (read: tx is greater than /?) is defined which has the following properties: (Gx) For tx, ft, y e A, {a) . Since all linear forms a 6 [£(?7)]j can be obtained in this way, we can identify the topological dual [E(U)]'b with the Banach space E'(U°) constructed in 0.11.2. 0.11.4. If A is an arbitrary set in 33(£), then for each linear form a e E' a linear form a € [£(¿1)]' is defined b y the equation = < x ay

for x € E(A) ,

where the norm of a is given b y P'ifi) = sup {|\ : x € A} = pA.(a) =

.

The normed space E'(A°) formed in the sense of 0.11.1 can t h u s be considered as a linear subspace of the topological dual [E{A)]'b b y identifying the equivalence class a(Aa) with the linear form a. Moreover, the normed space E(A) can also be considered as a linear subspace of the topological dual [E'(A°)~]'b. For this we must associate with each element % e E{A) the linear form g e [E'(Aa)Y, which is defined b y the equation